Sources of uncertainty in a DVM-based measurement system for a
quantized Hall resistance standardVolume 99, Number 3. May June
1994
Journal of Research of the Nationai Institute of Standards and
Technology
[J. Res. Nail. InsL Stand. Technol. 99, 227 (1993)]
Sources of Uncertainty in a DVM-Based Measurement System for a
Quantized
Hall Resistance Standard
Kevin C. Lee, Marvin E. Cage, and Patrick S. Rowe
National Institute of Standards and Technology, Gaithersburg, MD
20899-0001
TVansporUible lO IcH standard rcstslors h<[ve become fairly
widespread in industrial, university, and government standards
labo- ralories because of iheir \tr*/ tempcralurc coefficieni of
resistance, ease of iransporta- Ijon. and convenient value. The
values of these resistors, however, tend to drift with lime,
requiring jieriodic recalibration against an inv-iu^iant standard
such as the quantized Hali resistance. The availability of a
simple, inexpensive measuremeni sys- tem for calibrating 10 kft
resistors against such an invariant standard would be of great
benefit to primary standards tab- oratories. This paper describes a
simple au- tomated measuremeru system that uses a single, high
accuracy, commercially avail- able digital voltmeter (DVM) to
compare the voltages developed across a 10 kil stan- dard resistor
and a quantized Hall resistor
when the same current is passed through the two devices. From these
measure- ments, the value of the 10 kQ standard re- sistor is
determined. The sources of un- certainty in this system arc
analyzed in detail and it is shown thai it is possible to perform
calibraltons with lelative com- bined standard uncertainties less
than lXl(r'(0.1 ppm).
Key words: digital voltmeter. DVM method; electrical metrology:
electrical ref- cicnce standards; quantized Hall resis- tance;
quantum Hall effect; resistance cali- bration.
Accepted: March 3. 1994
1. Introduction Resistors composed of coils of wire wound
around
suitable forms have been used as staixiards of resistance for many
years [IJ. Such devices arc even tod^ widely used as worlcing
standards of resistance by primary and secondary standards
laboratories in industry, university, and government. Due to aging
of the wire and other effects, however, the values of these
resistors tend to drift with time, requiring periodic recalibration
against a known standard. Because wire-wound resistors drift with
time, many national standards laboratories have adopted a standard
of resistance based on the quantum Hall effect [2]. When a sample
containing a thin, two dimensional conducting layer known as a
two-dimen- sional electron gas (2 DEG) is cooled to liquid
helium
temperatures in the presence of a very strong magnetic field, the
resistance of the device becomes quantized, assuming well defined
values given by
R»{i) VH(I) h_
ie' i (I)
where VH is the voltage across the Hall device. / is the current
through the device, h is the Planck constant, e is ihe elementary
charge, / is an integer, and R^, the von Klitzing constant, has
been defined by international agreement to be exactly 25 812.807 Cl
for the purposes of practical electrical metrology [3], The
resistance is
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Volume 99, Number 3, May-June 1994 Journal of Research of the
National Institute of Standards and Technology
time invariant, and is, under appropriate conditions of
measurement, independent of the measurement condi- tions, such as
current, temperature, and magnetic field. Because of these
properties, the quantized Hall resis- tance, by international
agreement, has been used since January 1,1990 as a practical
representation of the ohm.
The measurement systems in use at national standards laboratories
are quite complex and are capable of achieving relative combined
staixiard uncertainties [4] of I X10 * (or 0.01 ppm where 1 ppm -
1X10 *) or less [5]. Many government and industrial standards
laborato- ries do not need such small uncertainties in their work.
Indeed, the uncertainty required by many laboratories is
sufficiently large that the drifts in the values of wire- wound
artifact staiKlards are less than their measure- ment resolution
and are therefore ignored. Such labiwa- tories are well served by
wire-wound artifacts which can be sent periodically to national
standards laboratories to be calibrated or to participants in
NIST's Measurement Assurance Program, or MAP. Generally, the
relative combined uncertainty achieved in the MAP is abtxit 0.1 ppm
to 0.2 ppm.
Some laboratories, however, require smaller uncer- tainties. At
such levels, drifts of the values of wire- wound artifacts require
that they be frequently recali- brated. For these laboratories, the
availability of a simple and inexpensive invariant standard of
resistance would be of great benefit. The fairly recent introductbn
of high accuracy commercial digital voltmeters (DVMs) with 8 1/2
digit resolution has made it possible to con- ceive of a simple and
fairly inexpensive measurement system [6] that would enable
government and industrial standards laboratories to perform
calibrations of their wire-wound resistors directly against a
quantized Hall resistor with uncertainties of 0,1 ppm or
less.
Such a calibration s>'stem has three distinct parts to it: a
quantized Hall resistance device, a cryogenic system in which a
superconducting solenoid and the Hall device are cooled to liquid
helium temperatures, and a mea- surement system for comparing the
standard resistor to this quantized Hall resistor. While the
selection of the sample and cryogenic system are beyond the scope
of this paper, a few words must be said about them, for they
al'fcct the design of the measurement system [7]. The sample and
the cryogenic system must be such that the conditions for accurate
measurement of the Hall resistance are met [8]. The conditions
pertinent to thus discussion are that the plateaus in the Hall
voltage ex- tend over as broad a range of magnetic field as
possible, and that the voltage drop along the sample in the direc-
tion of the current flow, V^, be as small as possible utxler the
conditions of measurement.
While in theory any Hall plateau [any value of i in Eq. (1)] can be
used for resistance calibration, in practice, the plateaus
corresponding to higher values of /' (i > 4) tend, except in
unusual cases, to have values of Vj which are unacccptably large
for precision resistance calibrations [91. In general, the plateaus
corresponding to lower values of j (;=4,2,1) (X:cur at higher
magnetic fields, are broader, and have values of V^ that are small
enough to permit accurate resistance calibrations. The value of
magnetic field at which any given plateau oc- curs LS a function of
the electron concentration in the 2 DEC which in turn is a function
of the design of the sample: samples can be designed to exhibit the
j-4 plateau, for example, at very low fields of only a few tcsla,
or very high fields of 10 T or more. In choosing a sample design,
one must balance several factors: if the plateau occurs at lower
field, it will be more accessible with an inexpensive magnet and
cryogenic system, but it will be narrower, and possibly V^ may be
too large; on the other hand, if the plateau occurs at a high
magnetic field, the plateau will be broader, Kr vvill be smaller,
but the cryogenic system and magnet will have to be much larger,
and consequently more expensive. It is the opin- ion of the authors
that the optimum sample exhibits the /=4 plateau [R»{4)-6 453.201
75 il] in a magnetic field range of 4.5 T to 6,5 T, and the (=2
plateau between 9.0 T and 13.0 T. TTiese fields are easily
attainable with commercially available superconducting magnets. The
discussion of the measurement system in this paper therefore
assumes that the resistance of the Hall device, «H(0, will be 6
453.20175 O, 12 906,403 5 fi, or 25 812.807 fl, corresponding to
the i-4,2, or 1 plateau.
2. Description of Measurement System
This measurement system is shown schematically in Fig. 1. The
standard resistor to be calibrated (Rs) Ls placed in series with
the Hall device (/?») A constant current is passed through both
resistors, and the poten- tial difference across each resistor is
measured with the DVM. All measurements of resistance are
four-terminal measurements. The current source and DVM are con-
nected to the resistors using switches, so that the direc- tion of
current flow can be easily reversed, and the meter can be connected
to either of the resistors. The measure- ments are performed in the
order described by Marullo- Reedtz and Cage [5]. The potential drop
VR across the standard resistor /?s is first measured with the
current flowing in the "positive" direction. The current direc-
tion is then reversed, and the potential drop across the standard
resistor is measured again. These two
228
Volume 99. Number 3, May-June 1994
Journal of Research of the National Institute of StandanJs and
Technology
1 A
Current Source
B ^fj,
Fig. I, DVM-basfd mcastircmcnl system for comparing wire-wound
standard resistors (*,1 with a <)uantized Hall resistor (RH)-
AL, and Ki; arc the net Icjikagc resistances to ground of the
cahles. current source, resistors, and the resl of Ihe measurement
system, The contacl resistances are denoted by n , r,^. The voltage
across each resistor is mcasuned with ihc digital voltmeter (DVM).
The thermal voltages generated at the connections between the DVM
and the resistors are denoted by e«, and el,. Z is the internal
impedance of the DVM and Yc is the Norton equivalent admittaiKc of
the current source.
measurements are repeated in the reverse order, to give a set of
four values:
VR(+;), v^-i). v^-i), VR(+/). (2)
The DVM is then connected to the Hall device, and this same
sequence of four measurement.s is made on it twice. Finally, the
standard resistor is measured again. The entire sequence of
measurenienls is:
(1) VR(+/), V„(-/), VR(-/), V„(+/) -
(2) V„(+/), VH(-/), VH(-/), V„(+/)
(3) V„(+/), V„W). VH(-/), VH(+/)
(4) VR(+/), VR(-/), VR(-/), Vp(+/)
In all, a group of sixteen measurements is taken. The four
individual values of each of these sets is averaged
<VR.> (3a)
{VH2) (3b)
<v„,) (3c)
<VR4) (3d)
to yield a group of four values shown at the right. It will be
shown later that this eliminates the effects of thermal voltages
that are either constant or vary linearly with time. Finally, the
two voltage drops across the standard resistor, (VRI) and (VR4) are
averaged, as are the two measurements of the voltage drop across
the Hall resis- tor. <Vw> and (VH3>:
^^^.<M|0^_
(4a)
(4b)
The above measurement sequence is then repeated with the positions
of the Hall resistor and the standard resistor interchanged: i.e.,
the Hall resistor is connected in the top position of the circuit
in Fig. 1, and the standard
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Volume 99, Number 3, May-June 1994 Journal of Research of the
National Institute of Standards and Technology
resistor is placed in the bottom position. The average voltages in
Eqs. (4a) and (4b) are then computed for this interchanged
position, yielding {V^^ and (V»°^).
In theory, if the current through the two resistors is
constant,
(vr> (5)
In practice, the situation is complicated greatly by a large number
of factors, including thermally generated voltages, leakage
resistances, contact resistances, insta- bilities in the current
source, noise in the voltmeter, etc. It is shown in Sec. 3 that by
application of the measure- ment procedure outlined above, the
effects of several of these factors can be eliminated or minimized.
Other factors cannot be eliminated by design of the measure- ment
system, and their effects must be independently minimized. Some of
these factors contribute to the er- ror of the determination of the
value of the standard resistor. At this point it should be noted
that the value of the standard resistor R is usually expressed as a
deviation from its nominal value, denoted Rs^:
\ RT" J Rr"
This ratio can be expressed in terms of the voltage ratios in £q.
(5) as:
" (v„) Rr '
As a result, the correction factors and uncertainties in the ratio
of the voltages in Eq. (5) must be divided by the
ratio RH
in order to obtain the correctbn factors and
uncertainties in the deviation of the resLsttff from its nominal
value.
In this paper, relationships between the magnitudes of the vaiious
effects and the resulting error in the value of the standard
resistor are determined. Ft is shown in Sec. 3 that if certain
limits are placed on the magnitudes of the various systematic
effects, the magnitudes of the factors required to correct for
their effect on the value of the standard resistor are
significantly less than the uncer- tainty in the value of the
standard resistor due to random effects, and the corrections can be
neglected while still maintaining a relative combined standard
uncertainty of
0.1 ppm. In Sec. 4, the uncertainties due to random effects, such
as nonlinear drifts in the output of the current source and in the
gain and offset of the DVM, etc., arc estimated. Limits arc derived
for the maximum values of these factors to ensure that the relative
com- bined standard uncertainty in the value of the standard
resistor is of the order of O.i ppm. It is shown that an
uncertainty of 0.1 ppm or less can be obtained when calibrating
resistors with Rs as much as a factor of 4 different from the
quantized Hall resistance, /?K(/) [this holds true whether the i =
1, 2, or 4 plateau is used for R„ii), i.e., whether the value
of/?„(') is 25 812.807 H, 12 406.403 5 n, or 6 453.201 75 ill
3. Corrections Arising From Systematic Effects
Systematic effects that can contribute significantly to the error
of the determination of the value of the stan- dard resistor arc
associated with four main parts of the measurement system. These
are:
i) the wires, cables, and switches used to connect the Hall
resistor and the standard resistor.
ii) the Hall device and standard resistor;
iii) the current source used to supply the current thnxigh the
resistors; and
iv) the DVM used to measure the voltages across the two
resistors.
The magnitudes of the errors arising from each of these parts of
the mea.surement system are estimated in this section.
3.1. Measurement System
In the measurement system shown in Fig. 1, the Hall resistor and
the standard resistor to be calibrated are connected in scries, and
are connected, by means of cables and switches, lo a current
source. In principle, the same current flows through each resistor,
so the ratio of the voltage measured by the DVM across the standard
resistor and the Hall resistor should equal the ratio of the values
of the resistors. In practice, however, there are thermally
generated voltages in the wires, switch contacts, and variiws
connections in the circuit which cause the measured voltages to
differ from the actual voltage drops across the resistors. It is
shown in Sec, 3.1.1 that the averaging technique described in Eq.
(3) above eliminates the effect of thermal voltages that are either
constant or vary linearly with time. Variable
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Volume 99, Number 3. May-June 1994 Journal of Research of the
National Institute of Standards and Technology
contact resistances in the switches connecting the resis- tors to
the current source can cause variations in the current supplied by
the current source. In Sec. 3.1.2, an upper limit is derived for
the magnitude of the permissi- ble variations in the contact
resistances. Leakage cur- rents in the cables, switches, and other
components of the system can also result in errors in the
measurements of the voltages. The errors due to the leakage
resistances can be corrected for, as described in Sec. 3.1,3.
3.1.1 Thermal Electromotive Forces Thermally generated voltages in
the switch contacts, the wires, and the connections to the DVM
result in erroneous deter- minations of the voltage drops across
the resistors in the circuit. These thermal voltages are denoted by
e,i, and e^ in Fig, I, and arise when there are temperature varia-
tions between various parts of the circuit and when there are
junctions between dissimilar materials, such as in switches or at
solder coniKCtions. Thermal voltages are therefcB^ generated at the
connections between the cur- rent source and each resistor, and at
the connections between the DVM and each resistor. TTic thermal
voltages generated in the contacts to the current source add to the
current produced by the current scxirce. As is shown in the next
section, if the equivalent admittance of the current source (K in
Fig. 1) is bw enough, the current source will adjust its output to
maintain a con- stant current through the circuit, and these
thermal voltages have no effect on the measurement. Thermally
generated voltages in the contacts to the DVM, on the other hand,
are significant, and must be corrected for. Since the thermal
voltages are a function of the temper- ature differences in the
circuit, they do not change sign when the current is reversed.
Thus, if one averages the voltage across one of the resistors
measured with the current in one direction and the voltage across
the same resistor obtained with the current in the opposite direc-
tion, the thermal voltage docs not contribute to the aver age.
Specifically, if the Hall resistor is in the BOTTOM position of the
circuit of Fig.), the voltage drop across it with the current
flowing in one direction (which will be denoted as the "positive"
direction) is
vTw) -/ GL, + G„
+ e*. (7)
^ (+/) - +/ Gu + GH
+ e* , (6)
where GL," I //?LJ is the leakage conductance in parallel with the
Hall resistor (described in more detail in Sec. 3.13) and GH =1/^H
is the Hall conductance. With the current flowing in the opposite
direction (denoted as the "negative" directkin) one obtains
The average value of the voltage across the Hall resistor is then
independent of the thermal voltage:
(vT') = I [vr (+/) -vW' (-/)] - ^^^ (8)
Tt has been assumed that the magnitudes of the ther- mal voltages
are the same and independent of the current direction. Practically,
the current is reversed using switches, but the connections between
the resistor and the DVM are not broken when the direction of the
current is changed. Thus, the thermal voltages in the DVM contacts
should not change when the current is reversed, and the assumption
that the thermal voltages remain constant or vary linearly with
time during the measurements should be a gotxi one. It is also
important to note that it was assumed that the current dtx;s not
drift with time. This is discussed in more detail in Sec.
3,2.
3.1.2 Contact Resistances Contact resistances occur at all of the
junctions in the system, including solder connections, switches,
and other connectors in the circuit. In addition, the ohmic
contacts to the 2 DEG in the quantized Hall device also exhibit
contact resis- tance. Voltages develop across only those contacts
that have current flowing thr<iugh them. The effects of the
resistances of the contacts between the resistors and the DVM arc
therefore minimized by making 4-terminal measurements of the
voltages across each resistor, i.e., the voltage across a resistor
is measured between two terminals that are separate from the
terminals that carry the current.
The contact resistances TJ,, ..., r^,, between the resis- tors and
the current source shown in Fig. I change the total resistance of
the circuit, but if they are constant and reproducible, as would be
the resistances at solder con- nections, then it is apparent that
they have no effect on the measurement. The resistances of the
contacts in the switches connecting the resistors to the current
source may change, however, for the switch contacts are con-
stantly being opened and closed. If these contact resis- tances are
reproducible, then they also have no effect on the total
measurement. To see this, consider the case where the total contact
resistance for one current direc- tion always differs by an amount
hZ from that for the opposite current direction. If the equivalent
admittance of the current source is zero, then obviously this has
no effect. In practice, however, the admittance of the
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Volume 99, Number 3, May-June 1993
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cuiTcnt source is nonzero, and this difference in contact
resistance changes the total load impedance, and slightly affects
the current through the resistors by an amount S/L. In this case,
the average in Eq. (8) is multiplied by the factor
(-i^)- (9)
where /L is the current through the resistors. If the differ- encc
hZ in the contact resistances is reproducible, then the averages of
both the voltages across the Hall resistor and the standard
resistor are multiplied by this same factor, and it cancels when
ratios of the voltages are taken (as described in Sec. 2, and in
more detail in Sec. 3.1.3).
If the contact resistances of the switches vary in a raralom,
uncoirelated, and irreproduciblc manner every time they are opened
and closed, each voltage ha.s a different correction factor of the
form given by Eq. (9). These correction factors do not cancel when
voltage ratios are taken. As a result, it is necessary to keep
8/iy/i. of the order of 10"" or less to assure uncertainties of 1
part in 10' in ihe determination of the value of the standard
resistor. This requirement places a limit on the variation in
contact resistance SZ which depends on the admittance of the
current source: the smaller the admit- tance, the larger the
permissible variations in SZ. Specifically, if the admittance of
the current source is K^. it can be shown that (10]'
YcSZ [lHR»+!is)YA
(10)
If the admittance of the current source is about 50 jiS, /f„(j-4) -
6 453.201 75 fi, and ;?s-IO kO,
afL = 2.7 X lOr* SZ.
and SZ must be less than 0.36 mfl for dljli. to be less than 10^.
Ff a low admittance current source is used, SZ can be larger, but
in general, the contact resistances in the switches and contacts of
the circuit .shcxild be kept very small, and should be reproducible
to within a few tenths of a milliohm, in order for them not to
affect the accuracy of the mea.surcment. This should not be diffi-
cult if care is taken to ensure that all of the contact surfaces
are very clean and not covered with a thin oxide or a layer of
organic contaminant.
' TTie Current source is represented by ils Norttw equivalenl.
i.c.. an ideal curreni source In in parallel with an equivulenl
iniemal admii- taivx Y;. See Rcf. [10] for more mfonnation.
The problem posed by the resistances of the contacts to the
quantized Hall device Ls a vastly more subtle one. As described
above, the difference between the resis- tances of the current
carrying contacts to the 2 DEC with forward and reverse directions
of current must be fess than a few tenths of a millk^m. It is very
difficult to produce contacts to the 2 DEG with such small con-
tact resistances [11], but fortunately, the actual contact
resistances need not be this low: the important point is that the
contact resistances must be independent of the direction (and
magnitude) of the current. Nevertheless, it is neces.sary that the
current carrying contacts have con- tact resistances of less than
10 mO, or they generate substantial amounts of noLse which prevents
accurate resistance comparisons. One would think that the resis-
tance of the contacts used to measure the Hall potential are
unimportant, for no current flows through them. These contact
resistances, however, must also be in the range of a few milliohms,
for reasons which arc beyond the scope of this paper [7, 121.
3.1^ Leakage Resistances Leakage resistances arise from the
noninfinite resistance of the electrical insulations used in
constructing the system. As a result, the leakage resistance is
distributed throughout the sys- tem: (here are contributions from
the current source, the cables, the DVM, and even the standard
resistor and the wires leading to the quantized Hall resistor. The
leakage resistances from the high terminal of the current source,
the current reversal switch, the cables, the standard resis- tor,
and the DVM have been combined in the idealized "leakage
resistance" R^ shown in Fig, 1. The leakage resistances from the
low terminal of the current source, cables, etc., and the quantum
Hall resistor have been combined in the idealized "leakage
resistance" Ri_.. The circuit is grounded between the Hall resistor
and the standard resistor to minimize the effects of the leakage
resistance between the low terminal of the DVM and ground: the low
terminal of the DVM is always con- nected to this point throughout
the entire measurement sequence. The high terminal is alternately
connected to point A in Fig. 1 to measure the standard resistor, or
point B to measure the Hall resistor. If it can be assumed that the
leakage resistances /?L, and /JL, remain constant throughout the
measurement cycle, and that the Hall resistor arxl the standard
resistor have nominally the same values, it can be shown that the
error due to fcak- age resistance is eliminated by averaging the
ratio of the voltage across the standard resistor to the voltage
across the Hall resistor with the resistors in the standard config-
uratk>n shown in Fig. 1 and the same ratio obtained when the
positions of the resistors in the measurement circuit are
interchanged. With the resistors connected as shown in Fig. 1 (the
standard position), the average
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voltages across the resistors [as determined from Eq. (8)1
are:
<vr>- GL, + Gs
(11a)
01b)
where G —\/R for each of the resistors in Fig. 1. The ratio of
these two voltages is then
<vr> " GL, + Cs
-^(1+-^ ^). (12) Gs \ GH GS /
if it is assumed that the leakage resistances are inde- pendent of
the positions of the resistors, the positions of the Hall resistor
and the standard resistor can be usefully interchanged. In this
case, the Hall resistor is in parallel with Rt, and the standard
resistor is in parallel with R^^. The ratio of Eq. (12) is
then
tonstnicted current sources will have leakage resis- tances greater
than about 10'' H (GL < 10"'^ S). If/?s - 10 kfl, and /fH - 6
453.201 75 O or 12 906.403 5 fl. the correction to the value of the
standard resistor will be of the order of 0.{X)3 ppm, which is more
than an order of magnitude less than the uncertainty due to random
ef- fects, and can therefore be neglected. For resistance ratios of
4 or more, as would result from the comparison of a 6 453.201 75 H
and a 25 812.807 11 resistor, how- ever, the correction to the
value of the standard resistor can be as large as 0.02 ppm, which
is comparable to the uncertainly due to random effects and cannot
be ne- glected. In this case, a correction factor can be estimated
by measuring the leakage resistance between the point C in Fig. 1
and earth which will be approximately equal to I/(GL,+GL;) (in Fig.
1 point C is connected to earth, but for this leakage resistance
measurement this connection must be removed).
The uncertainty associated with the assessment of this leakage
resistance, however, will be quite large, possibly as large as the
conection factor itself, so one can treat this correction factor as
a component in the assessment of the combined uncertainty. If lower
uncer- tainties are required, however, it will be necessary to
increase the leakage resistance of the system another order of
magnimde, something which is quite difficult to do.
(vD GL,+GH a <Vff*> GL, + GS G ;('-|;-t)- '">
Averaging Eqs (12) and (13) yields:
2 l<vr)"- {v^-})
GLI + G|.,
2Cs (14)
If the standard resistor has exactly the same value as the Hall
resistor, then the last two terms on the right side of Eq. (14)
cancel, and the leakage resistances have no effect. If the
resistors are not equal in value, then there Is a small correction
factor. If the leakage resistances change when the current is
reversed or the resistors arc interchanged, then the above analysis
does not hold. Even in this ca.sc, however, the error in the
determina- tion of the value of the standard resistor caused by the
leakage resistances will be of the order of the ratio of the
leakage conductance CL=GI ,+Gi.j to the conductance of the larger
of the two resistances K,i or Rs- Typically, cables insulated with
PTFE Teflon, and carefully
3.2. Current Source
Several factors determine the optimum current for these
measurements. The higher the current, the larger the voltages
across the resistors, and the smaller the averaging time required
to obtain voltage measurements with a given experimental .standard
deviation of the mean. Too large a current, on the other hand, can
cause self-heating of the standard resistor, which will change its
value and, more importantly, can cause breakdown of the
dissipationlcss current flow in the Hall device, rcn dcring it
unusable for resistance calibrations [ 13J. Typi- cally, currents
between 10 jjiA and 50 p.A satisfy these conditions. It should be
noted that when performing calibratk>ns of 10 kXl resi.slors
against a quantized Hall resistor, it is the maximum current that
the quantized Hall device can sustain withcHit breaking down that
lim- its the maximum current at which mca.surements can be
performed. Generally, this critical current is far below the
current at which even a typical 10 kil resistor would start to show
self-heating effects.
It should be recognized, however, that a reduction in averaging
time due to a higher current is only realized if the primary factor
limiting the accuracy of the voltage measurement is the
signal-to-nolsc ratio and not the resolution of the meter. If the
primary limiting factor is
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Volume 99, Number 3, May-June 1994 Journal of Research of the
National Institute of Standards and Technology
the resolutbn of the meter, then the benefits of increas- ing the
current are somewhat limited. For example, if the resolution of the
meter is 10 nV on the 100 mV range, then one benefits from choosing
a current that produces a voltage near the top of this range, as in
this case the voltage measurement has the minimum relative
uncertainty. Increasing the current so that the voltage is so large
that the meter must use the next range, e.g., the 1 V range,
however, may not result in any benefits if the resolution of the 1
V range is 100 nV. In this case, the relative uncertainty of the
voltage measurement will be the same or worse than at the lower
current. In practice, some benefit may be obtained by using the 1 V
range in this example, tor the internal DVM noise is often less on
the higher voltage ranges than on the 100 mV range, so lower
uncertainties in the measurement of the voltage can be achieved
with shorter measurement times, even though the resolution of the
meter is poorer on the 1 V range.
The method for determining the ratio of the value of the standard
resistor to the Hall resistance described in Sec. 2 assumes that
the current through the resistors is constant during the time that
the voltage measurements are made. Any variations in the current,
such as drifts or noise, will appear as drifts or noise in the
voltages across the resistors and will affect the accuracy of the
calibration of the standard resistor. In order to obtain a reladve
combined standard uncertainty in the calibration of 10"\ it is
necessary to keep the noise in the current source aixl nonlinear
deviations in the current below 0.01 ppm; for currents between 10
fj,A and 50 jiA, this dictates that the current variations be less
than 0.1 pA to 0.5 pA. Such low noise levels are rather difficult
to achieve with active current sources containing opera- tional
amplifiers, transistors, or other solid state compo ncnts which
usually have noise levels of the order of a few ppm, and are
therefore generally unacceptable for use with this method. Current
sources made using mer- cury batteries and current-limiting
wire-wound resistors are capable of meeting these stringent
requirements, even though the output of these current sources tends
to decrease with time in a predictable, linear manner.
Fortunately, .such stable linear drifts in current do not affect
the measurements if the measurement sequence described in Sec. 2 is
used. If the current decreases at a constant rate B, and the
current at the beginning of the first measurement is /, then the
current at a time r after the first measurement was begun is
/(0 = I-S^ (15)
each voltage measurement in the group is given by the expression in
colunm 2 of Table 1. As described above in Sec. 2, the first four
voltage measurements (of the standard resistor with positive and
negative current) are averaged to eliminate thermal voltages,
yielding the value shown in the third column of Table 1. The
average voltages across the standard resistor are then averaged, as
are the average voltages across the Hall device, as described in
Sec. 2, Eq. (4a), to give
(vp, . (iiiliiiW . «. (^-)
<vr> = <VH2) + <Vm) «„[,,-(f«i,)]
If each individual voltage measurement described in Sec. 2 takes a
time Ar, then the current at the start of
(16b)
The effective current is the .same in both of these equations, and
is eliminated when the ratio of VH to VR
is taken. Note that the effective current in both of these
equations is the current at the exact mid-point of the measurement
cycle.
33 Digital Vtdtmeter (DVM)
The quality of the digital voltmeter used to measure the voltages
across the resistors is the ultimate factor limiting the accuracy
of this technique. In order to ob- tain resistance calibrations
with relative combined stan- dard uncertainties of 10"\ the DVM
must be capable of measuring voltages with uncertainties about a
factor of 5^10 less than this. If a 20 JJLA current is used, the
voltages across the resistors will be of the order of 200 mV, and
the DVM must be capable of resolving voltages of the order of 0.2
X10 "^ V, or 20 nV [14]. Commer- cial DVMs arc now available from
several manufactur- ers that have such a high resolution. In
addition to the high resolution, however, the DVM must have very
high accuracy, a high degree of linearity, a high input impedance,
high stability, and very low noise. Commercial 8 1/2 digit
multimeters from several manu- facturers arc on the market that
meet these specifica- tk>ns. In this section, various systematic
effects
234
Volume 99. Number 3, May-June 1994
Journal of Research of the National Institute of Standards and
Technology
IfaMe I. Assuming a constani rate of change in the currcnl
pro(tu<K<l by the current sounre and thai each ^^lage
mcasurcincnt lakes a time A/, the current is calculated at the
beginning of each voltage measure- ment on the standard resistor
(R^) and and the Hall resistor (AH)- Each set of four measurements
on a resistor is averaged to eliminate thermal voltages and other
constant offsets, resulting in the average voltages shown in column
3.
Resistor Mean voltage
/fH(+) +(/(r-llBAf)
«s[/^(f)BA<]
/fK[/n-(-YjBA/j
>?H[/t--(y)sArj
associated with the DVM that contribute to the total uncertainty
arc analyzed. In Sec. 3.3.1, a correction tenn accounting for
offsets and nonlinearities in the response of the DVM to applied
voltages is deriwd. In Sec. 3.3.2, the effect of the small current
source between the input terminals of the DVM is con.sidered. The
ef- fect of noninfinite input impedance is considered in Sec.
3.3.3.
While this measurement system is quite similar to potentiometric
measurement systems achieving smaller ultimate uncertainties [5],
the accuracy, range, resolu- tion, and linearity requirements on
the DVM used with this system are greater. In the potentiometric
measure- ment systems, a potentiometer is used to cancel the Hall
voltage, so the detector is only used to measure very small
differences between the Hall voltage and the voltage drop across
the standard resistor being cali- brated. Therefore, the detector
need not have a very great range, but must have very low noise,
high resolu- tion, and high accuracy. Furthermore, since the
detector is only measuring small deviations from zero, the linear
ity of the detector over large ranges is not crucial. In the
measurement system described in this paper, the DVM is used to
measure voltages that differ widely from zero, and that are both
positive and negative. This requires the DVM to have a very high
degree of linearity.
Offsets and nonlinear responses of the DVM can be determined by
calibrating the DVM against a Josephson airay. The Josephson array
produces a time-invariant voltage that is related to fundamental
constants, and, by international agreement, provides a practical
metroJogi- cal represcntaiit>n of the volt. If the Josephson
array produces a defined voltage V, the voltage indicated by the
DVM will be:
VDVM = A + gr + A^(T), (17)
where A is the offset, g is the gain of the DVM, and N(y) is a
nonlinear correction. For most modern high quality meters,
g - 1 + 5. (18)
where 5 is a small number. The values of A, ^, and N should be
determined by measuring VDVM with applied array voltages in the
neighborhood of the values ex- pected to be encountered in the
resistance measure- ments. A least-squares fitting procedure should
be used to determine the gain, offset, and nonlinear corrections
for both positive and negative voltages.
If the offset voltage A is the same for both positive and negative
voltages, then the same averaging proce- dure that eliminates the
thermal voltages will eliminate the offset voltage: the offset will
cancel when V'(-(-/) and V{-/) are subtracted, as in Eq. (8). In
practice, however, neither A nor g need be the same for positive
and nega- tive voltages.
As described in Sec. 2, the ratio of the resistor values is
determined from the arithmetic mean of the ratios of the averaged
voltages across each resistor in the standard and interchanged
posilbn, as in Eq. (14). Each voltage in Eq. (14) must be corrected
for the offset, nonunity gain, and nonlinearities in the DVM before
the ratios are taken. A factor taking all of these corrections into
ac- count can be derived for Eq. (14) as follows: the voltage
indicated by the DVM when it is connected to the Hall resistor in
the bottom position in the circuit of Fig. 1, with positive
current, is given by (neglecting thermal voltages)
V^(+/) = A^ + S*Tr + N(rr), (19)
where "VH^ is the' 'true" voltage across the Hall resi.stor in the
bottom position in the circuit of Fig. 1. Likewise, with the
current reversed.
vr(-/)=A- - g- rr+N{-.rry (20)
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Vblume 99, Number 3, Mj«y-June (994 Journal of Research of the
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Averaging as in Eq. (8) gives:
A*-A where
SN^~-^ ^-^ ^. (2ld)
If leakage resistances and other systematic effects are ignored,
the ratio of the "true" voltages across the resistors is equal to
the ratio of the resistors, i.e..
Inverting Eq. (21a) above gives
rr- (VD -a -SNfT
(22)
(23)
Repeating the calculation for each of the voltages and substituting
into the left hand side of Eq. (22) yields
(W y-a-SNT
(24)
Generally, the corrections a and 5JV are very much smaller than the
values of the voltages (V), so that the ratio of a+8N to (V) is of
such small magnitude that terms of second order in this quantity
can be neglected. Equation (24) can Chen be simplified to
^"2 l(vr) (vn^
({vf^\ ( « + &vr . a + mT VvT) )(- <vf^ <vr) )]
(25)
The first term on the right side of Eq. (25) is the ratio of the
voltages across each resistor [averaged as in Eq. (3)], while the
second term on the right side of Eq. (25) is the correction factor
that must be applied to correct for nonzero offsets and
nonlinearities in the DVM. This expression can be considerably
simplified by noting that both a and dN are very much smaller than
the voltages (V), so the denominators can be replaced by their nom-
inal values. ]n addition, the ^ are the differences between the
nonlinearity corrections for positive and negative voltages, and
since the magnitudes of the voltages across a given resistor are
alt essentially the same, little error will be incurred by making
the approx- imation that SN is a constant for each resistor. Thus,
with the approximations
<vr') - <vn - {Vu)
^(^(«-HM^„)-(a.6Ar.)). (27)
It is important to remember that a is die difference between the
offsets for positive and negative voltages, as given by Eq. (21)
above. Usually, the offsets are negli- gibly small and are the same
for positive and negative voltages, so that a in die above equation
can be ne- glected. In this case, Eq. (27) can be written
The quantity 8N in the above equation is the difference between the
nonlinearity correction for positive voltage and for negative
voltage. This need not be zero and must be determined by
calibrating the meter against a Joseph- son array. It should be
noted that if both resistors have nominally the same value, then
even though bN^ and ?>N„ may not be zero, Ihcy will have
essentially the same value, and the correction term in Eq. (28)
will vanish.
236
Volume 98, Number 3. May-June 1994
Journal of Research of the National Institute of Standards and
Technology
Corrections for the non linearity of the meter therefore need only
be made in the case that the resistors being compared have
nominally different values. In the case that the ratio of the
resistances is 4, this correction term can be quite significant.
Using the DVM calibration data in Fig. 2 of Cage et al. [6], and
assuming Rs-25 812.807 il and /?H-6 453.201 75 «, the correc- tion
terin of Eq. (28) would be 0.40 ppm ± 0.28 ppm. It is important to
note that the correction to the value of the resistor /?s,
expressed as a relative deviation from its nominal value,
(/?s-A?s°"y/^, due to nonlinearitics in the DVM is obtained by
dividing Eq. (28) by the ratio i?s™//?H (sec Sec. 2), which in the
case of this example is 4. The correction to the value of
(Rs-Rs^}/Rr" due to the DVM nonlinearily is therefore only O.I ppm
± 0.07 ppm in this example.
Since the nonlinearity correction term above may change with time,
the DVM should be calibrated before each resistaiKe calibration is
performed. It should be noted that if either the offsets or the
gain vary randomly with time, they will not cancel in the above
equations, and will give rise to an uncertainty in the final
determi- nation of the value of the standard resistor as discussed
in Sec. 4,
3.3^ Internal Current Source Modern digital voltmeters inject a
small current into the circuit to which their input terminals arc
attached. This current is often very small: for a Hewlett Packard
3458A DVM^ it was measured to be 10^"* A (with no bias applied to
the meter input terminals) by connecting a Keithley 602
electrometer directly across the input terminals. If this current
does not change sign or magnitude with varying applied bias, it
should have no effect on the average voltage measurements, as any
offset produced by it wouki cancel when the voltages measured with
positive and negative current are averaged. If it changes sign and
magnitude with changing applied bias, but generally retains a
magnitude in the range of lOr'* A, and if the measurement current
is in the range of 10 JLA to 50 n,A, this small current would
result in an error in the ratio of the resistors of only 0.0002 ppm
to O.OOlO ppm.
3.3J Input Impedance Whenever a DVM with finite input impedance is
connected across one of the resistors, some current will be shunted
through the DVM. If the input impedance of the DVM is ZDVM=
l/GuvM, and if the admittance of the current source is negligibly
small (that is, the current source is assumed to
be ideal), then the correction to the resistance ratio with the
resistors in the standard pc)sition will be:
' Certain commercial cquipmem, instmmems. or materials arc idemi-
fied in ihis paper in order to adequately specify the experimental
piDCfdure. Such idenlificaiion does not imply reeominendaiion or
endorsemcni by the National Institute of Standards and Technology,
nor dors it imply thai (he materials or equipment identified are
neces- sarily ihc best available for the purpose.
<vr> GH + GDVM
^H \ 2DVM / (29)
In the event that the two resistors being compared are of nominally
equal value, the correction factor in Eq. (29) will vanish. If
however, the resistors have different nominal values, the factor
must be evaluated, and can be quite appreciable, particularly if
the input impedance of the DVM is less than 10" 11. In this case,
if /?s-/rH((=l)=25 812.807 H and «H('-4)«6 453.201 75 n (a 4-to-l
resistance ratio) the correction to the value of Rs will be as much
as 0.019 ppm. In practice, the input impedaiKes of modern DVMs are
somewhat higher than 10" fl. The input impedances of two Hewlett
Packard 3458A DVMs were measured using a method described by Cage
et al. [6]. The impedances, measured at 22 °C with a relative
humidity of 43%, were 5.9X10'" D. for one DVM and 3.6xlO'^n for the
other DVM. Such input impedances would lead to only a 0.001 ppm
correction to the resbtance ratio when calibrating a 10 kCt
resistor against /f„(i_4)-6 453.201 75 ft, and a 0.005 ppm
collection lo the resistance ratio when calibrating a 10 kft
resisted against R»{i-\)- 25 812.807 ft.
4. Uncertainties Arising from Random Effects
The discussions in the previous sections have con- cerned only
errors and sources of uncertainty in the determination of the value
of the standard resistor due to systematic effects. As can be seen
from the summary in Table 2, the errors associated with most of
these system- atic effects can be reduced to the level of 0.0] ppm
by appropriate design of the measurement system. Ran- dom
variations in the various parts of the measurement system, such as
nonlinear drifts in the current, thermal voltages, contact
resistances, and even the offset and gain of the DVM can also
contribute to the uncertainty in the determination of the value of
the standard resistor. Many of these have been discussed in Sec. 3
aixl their effects are summarized in Table 3. Appropriate design
and construction of the measurement system, as noted in Sec. 3,
generally results in contributions to the com- bined standard
uncertainty from these effects of a few parts in 10*. External
noise picked up by the cables in
237
Volume 99, Number 3. May-June 1994
Journal of Research of the National Institute of Standards and
Technology
the measurement system, and circulating currents aris- ing from
multiple ground connections in the circuit (commonly referred to as
"ground loops") can also contribute to the combined standard
uncertainty. Fortu- nately, the influence of these effects on the
measure- ment can be greatly reduced by adequately shielding the
measurement cables and equipment, preferably with two layers of
shielding, and checking the circuit to ensure that there is only a
single connection to ground (sec Ref. [5J). Often this latter task
is complicated considerably by the fact that some instruments have
various internal connections to ground that are not obvious, so
that while
TbUe 2. Systematic effects thai give rise to coircciions lo the
value of the standard resistor dctemuned using this method. If the
value of each effect is kept within the limits shcFwn in the second
cdumn, the value of the correction lo the deviaiicm of the value of
Rs from its nominal value, given by the quantity (R^-Rf"VK%^. will
be shown in the thiid ct^umn. The fiiS value was calculated
assuming that lfs-10 kn and Rt,(i-4}-6 453.201 75 fl; the second
value was calcu- lated assuming As-10 kO and Rn-25 812.807 n. Since
the values of these corrections are generally very small in
comparison with the uncertainty due to random effects, these
correction factors are not applied, but for simplicity arc included
in the uncertainty of the final result.
Effect
Afttuc to result calculated: (in ppm)
leakage resistance >10''n 0.004 to 0.016 3.1.3
DVM input impedance >10"n <0.003 to 0.015 1.3 J
DVM nonlincarity <0-05 n-V Oio<0.10*0.07 3.3.1 ±0.01
|iV
IWble 3. Randomly varying effects that com ributc to the
uncertainty in the determination of the value of the standard
resistor. In the first row, the first number in the third column
was calculated assuming Ks-IO kfl and RH-6 453.201 75 fl; the
second number was calculated assuming flH-25 812.807 il.
Effect Value Standard Section in uncertainty which
(in ppm) calculated
DVM current source <10r"A 0.000 to O.OOI 3.3.2
Noise in DVM 5nV to 15 nV 0.015 lo 0.046 4 for 30 s for single
group of
measurements measuremenis
it may appear that the meter is isolated from ground, it in fact is
cither connected internally to ground through its power cable, or
through the chassis, which may be screwed to a rack, which itself
may be groutxled.
Even if the effects of external noLse, random drifts in the system,
etc., can be eliminated, the fundamental fac- tor limiting the
accuracy of this technique is the internal noise and resolution of
the DVM. Modern commercial 8 1/2 digit multimeters are available
that have resolu- tions as fine as 10 nV out of 1 V, as mentioned
in Sec. 3.3, so the DVM resolution does not really limit the
accuracy of this technique. The internal noise in the DVM, however,
can have rms values as high as 0,04 ppm to 0.1 ppm. TTiLs, coupled
with random, long-term drifts in the gain, offset, and
nonlinearity, are primarily responsible for limiting the accuracy
of this technique.
In order to minimize the effect of such noise on the measurement,
we must first examine the nature of the noise and how it affects
the measurement. What we measure are the voltages across the
resistors. These may be regarded as fixed voltages, upon which are
super-im- posed noise voltages, which can have different frequen-
CKS and amplitudes. Very high frequency noise, with a mean period
less than the time required for a single measurement, will not
greatly affect the measurement, for it will average to zero during
the measurement time. Thus, the uncertainties in the voltage
measurements in Eq. (3) can be greatly reduced by averaging each
voltage measurement for a bng time. This can be done by making
repeated measurements of each voltage over an extended period of
time and then averaging these mea- surements to produce a mean
voltage. The uncertainty in this mean will be given by [see Ref.
14]
5 = N{N - 1)
(30)
It would appear from Eq. (30) that the more measure- ments of each
voltage are made, the smaller the uncer- tainty in the mean. This,
however, is not the case, for in addition to the obvious
high-frequency noise that gives rise to scatter in the individual
measurements (each of which are assumed to be made over a short
time interval of a second or less), there are long term drifts in
the current, thermal voltages, and components of the mea- surement
system. The.se drifts can be considered to be "noise" with very low
frequency. In addition, there may be long-term, predictable drifts
in the system: an example of this would be the stow, linear
decrease in current produced by a mercury battery-powered current
.source as the batteries get depleted. Averaging a voltage
measurement for longer periods of time will decrease
238
Volume 99. Number 3. May-June 1994 Journal of Research of the
National Institute of Standards and Technology
the effects of higher frequency noise wiih periods less than the
measurement time, but if these long-term linear drifts in current
(for example) are present, there is some measurement time beyond
which the effects of short term noise become negligible, and the
primary coniribu tion to the uncertainty given by Eq. (30) will be
the linear drift in measurement current. For a drift of atxxit 0.2
ppm in 10 min, this time can be as short a.s 10 h or as long as
several days, depending on the magnitude of the noise
voltage.
For optimum results, the individual voltage measure- ments
indicated in Eq, (3) should be averaged for short enough times that
long term drifts do not contribute to the uncertainty of the
measurement at ail. Then, as was shown in Sec. 3.2, appropriate
averaging of sets of such measurements causes the effect of the
long-term drifts to vanish. With modern 8 1/2 digit DVMs, averaging
each of the voltage measurements in Eq. (3) for a period of about a
minute results in uncertainties in the determi- nation of the mean
value of each voltage of between 5 nV and 15 nV, depending on the
internal noise of the DVM. This results in a relative standard
uncertainty in the determination of the resistance ratio from a
single group of measurements of between 0.015 ppm and 0.046
ppm.
It is important to understand that this estimate of the uncertainty
in the resistance ratio oidy considers the contributions from the
noise, and does nor include other random effects, such as
irrcproducible contact resis- tances, and various random drifts in
the system. There- fore, while the uncertainty due to noise m^ be
quite small, the actual value of the ratio may be considerably
different from the value determined from a single group of
measurements- In order to reduce the combined un- certainty due to
all raitdom effects in the determination of the resistance ratio,
it is therefore necessary to per form numerous groups of
measurements, and then aver- age the resistance ratios r determined
from each group to obtain a final average < n >:
1 "' (31)
Because long-term drifts and other random effects tend to cause
fairly large fluctuations in the values of the r^ determined from
different groups (usually larger than the uncertainty in each r,
due to noise), the uncertainty in the final resistance ratio
determined from Eq. (31) is determined from the set of group means
< r^ >
N{N-i) ^{r.-<r>f (32)
The uncertainty calculated from Eq, (32) includes the effects of
noise, and of random variations in the system, so there is no need
to include these effects explicitly. The magnitude of the
uncertainty will depend on the magnitudes of these various random
effects. When the internal noise in die DVM is sufficiently low
that each voltage measurement has an uncertainty on the order of 5
nV to 7 nV. noise in the DVM will actually be only a minor factor
contributing to the final uncertainty: more significant
contributions will come from the irrepro- ducibility of the contact
resistances and other factors shown in Table 3. If these random
effects are kept within the limits shown in Table 3, however, it is
usually possi- ble to obtain relative uncertainties (due to random
ef- fects) in the determination of the mean resistance ratio [Eq.
(31)1 of less dian 0.01 ppm after 20 groups of measurements.
5. Conclusion Starvlards laboratories requiring resistance
calibra-
tions with relative combined standard uncertainties of less than
0.1 ppm could benefit from a simple, inexpen- sive, intrinsic
resistance standard. While the quantum Hall effect provides such a
standard of resistance, the measurement systems used at most
national standards laboratories are far too expensive, complex, and
time consuming to construct and use in government and in- dustrial
standards laboratories. This paper has analyzed the sources of
uncertainty arising from both systematic and random effects in a
simple quantized Hall resistance measurement system that uses a
modern commercially available 8 1/2 digit digital voltmeter to
compare the voltages developed across a standard resistor placed in
series with a quantized Hall resistor. A measurement sequence has
been presented which minimizes the ef- fects of thermal voltages
and linear drifts in the current on the final determination of the
unknown resistance. Criteria have also been presented for
minimizing the effects of the contact resistances and leakage
resis- tances.
Table 2 summarizes the systematic effects that cause errors in the
determination of the resistance using this technique. Most of these
errors are so small as to be negligible in comparison with the
uncertainty in the final resistance ratio due to random effects,
aixi these corrections arc therefore neglected and for simplicity
are included in the uncertainty of the final result. In the case of
the nonlinearity of the DVM, however, the error can be fairly
large, and a correction factor [derived in Eq. (25)J must be
applied to the final result. Table 3 summarizes the pcrmi,ssible
magnitudes of random vari- ations and drifts in the various
components of the mea- surement system.
239
Volume 99, Number 3, May-June 1994
Journal of Research of the National histitute of Standards and
Technology
The combined standard uncertainty in the value of die standard
resistor is the square root of the sum of the squares of the
standard uncertainties arising from both random and systematic
effects in the voltage measure- ments. If the uncertainty from
random effects is less than 0.06 ppni (easily achievable with just
a single group of measurements), the relative combined standard un-
certainty in the determination of the resistance can be as low as
0.06 ppm if the DVM nonlinearity correction in Table 2 is
negligible (including for simplicity the correc- tion factors in
Table 2 as sources of uncertainty). If the DVM nonlinearity
correctbn is as high as that shown in Table 2, the combined
uncertainty will be as high as 0.09 ppm.
If the uncertainly due to random effects is less than 0.01 ppm,
which can be achieved by averaging up to 20 groups of data as
described in Sec. 4, the relative com- bined standard uncertainty
in the value of the standard resistor can be less than 0.03 ppm
(again assuming the DVM nonlinearity correction to be negligible,
as waukl be the case if the resistors being compared had the same
nominal value). This DVM-based measurement system can, therefore,
be used to compare wire-wound standard reference resistors with
quantum Hall resistors with a relative combined uncertainty of less
than 0.1 ppm, and in the most favorable cases, with uncertainties
less than 0.06 ppm. Since the quantized Hall resistance does not
drift with time, this measurement system can be used to calibrate
wire-wound resistors, the values of which tend to drift with
time.
6. Acknowledgements
Tlie authors gratefully acknowledge the support of the Calibration
Coordination Group of the Etepartment of Defense for this work.
They also wish to thank Drs. B. F. Field, B. N. Taylor, R. F.
Dziuba. and A. F. Clark of NIST and Prof. C. A. Lee of Cornell
University for helpful discussions.
guantizeri Hail Resistance, J. Res. Natl. Bur. Stand. (U.S.) 92,
303^310 (1987).
[6] M. E. Cage, D. Y. Yu. B. M. Jeckelmann, R. l. Steiner, and R.
V. Duncan, Investigating (he Use of Multimeters to Measure
Quantized Hall Resistance Standards. IEEE TYans, Instnim. Meas. 40.
262-266(1991).
[7] For more information on the selecticm of samples, see W, van
dcr V/e\. R G. Haajiappe). J. E. Mooij. C. J. P. M. Hannans, J. P.
Andrf. G. Weimann. K, Ploog, C.T. Foxon, and JJ. Harris, Selection
Criteria for AlGaAs-GaAs Hetcrostmciures in View of Their Use as a
(Juanlum Hall Resistance Standard, J. Appl, Phys. (SS. 3487-3497
(1989).
[8] F, Dclah^e, Ttchnical Guidelines for Reliable Measurements of
the Quantized Hall Resistance, Metrolc^ia 26, 63-68 (1989).
[9] See, for example Fig. 6 in K. Jaeger, P. D. Levjnc, and Cj\.
Zack. Industrial Experience with a Quantized Halt Effect Sys- tem.
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[101 P. Horowiu and W. Hill, The Art of Electronics (Cambridge
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[Ill D. Jucknischke,H. J. Buhlmann. R. Houdi^. M. Itcgcms, M. A.
Py, B. Jeckelmann, and W. Schwiiz, Properties of Alkiyed Au- GeNi
Contacts on GaAs/GaAIAs HetetDstnsclures. IEEE Trans. Instrum.
Meas. 40, 228-230 (1990),
(12} H. Hirai and S. Komiyama, A Comacl Limited decision of the
Quantized Hall Resistance, J. Appl. Phys. 69, 655-662 (1990).
(13) M. E. Cage. R. F. Dziuba. B. F. Field, E. R. WHUams. S. M.
Girvin. A. C. Gossard. D. C.Tsui, and R, J. Wagner, Dissipation and
Dynamic Nonlinear Behavior in the QusUilum Hail Regime, Phys. Rev.
Lett. SI, 1374-1377 (1983).
114] The uTKertainly of the DVM measttfement is 1/2 V3 of its
resolution. The derivation of this result is given in Aruiex F2 of
the Guide to the Expression of Uncertainty in MeasurettKiU
(Intemaiiona) Organization for Standardizalion. Geneva,
Switzerland, 1993). p. 53.
About the authors: Kevin C. Lee and Marvin E. Cage are physicists
in the Electricity Division at NIST. Patrick S. Rawe, a student
from the Massachusetts Institute of Technology, was a Guest
Researcher in the Electricity Division when this work was done. The
National Insti- tute of Standards and Technology is an agency of
the Technology Administration, US Department of Com- merce.
7. References
|5|
See, for example. J, L, Thomas, Precision Resistors and their
Measurement, in Precision Mcasurcmcni and Calibration: Elec-
tricity and Elecirtmics, Natl. Bur. Stand. (U.S.) Handbook 77, Vd.
1(1961), pp. 111-142. K. v. Ktiizing, G. Dorda. and M. Ftpper. New
Method for High-Accuracy Deicrraination of the Fine-Structure
Ctwisiam Based on Quantized Hall Resistance. Phys. Rev. L^ii. 45,
494- 497(1980). BIPM Proc,-Vert). Com, Im. Poids ei Mcsurcs 56. 20
(1988). B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluating
and Expressing the Uncertainty of NIST Mcasurctnent Results. Natl.
Inst. Stand. Technol. Itch. Note 1297 (January 1993). See. for
example, G, Manillo-Recdtz and M, E. Cage. An Auto mated
Poteniiomciric System for Precision Measurement of the
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